Abstract

The Maréchal approximation for the Strehl ratio versus rms wavefront distortion is widely used in atmospheric scaling law codes, but a complete derivation is absent in the open literature. I present an updated derivation of the first term in Maréchal, then a complete derivation. The Strehl ratio is proportional to the square of the Fourier transform of the probability density function of the random phase noise. For Gaussian noise, the traditional Maréchal formulation is the result. The more general formulation suggests a method for characterization of random phase aberrations.

© 2009 Optical Society of America

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  1. K. Strehl, “Über Luftschlieren und Zonenfehler,” Z. Instrumentenkd. 22, 213-217 (1902).
  2. L. C. Roberts, Jr.,, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy “Is that really your Strehl ratio?” Proc. SPIE 5490, 504-515 (2004).
  3. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1997), Chap. 9.1.3, Eq. (24), p. 464.
  4. A. Maréchal, “Étude des effets combinés de la diffraction et des aberrations géometriques sur l'image d'un point lumineux,” Rev. Opt. Theor. Instrum. 26,257-277 (1947)
  5. G. Martial, “Strehl ratio and aberration balancing,” J. Opt. Soc. Am. A 1, 164-170(1991); see comment surrounding Eq. (8).
  6. V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am. 72, 1258-1266 (1982).
    [CrossRef]
  7. D. D. Lowenthal, “Maréchal intensity criteria modified for Gaussian beams,” Appl. Opt. 13, 2126-2132 (1974).
    [CrossRef] [PubMed]
  8. V. N. Mahajan, “Strehl ratio of a Gaussian beam,” J. Opt. Soc. Am. A 22, 1824-1833 (2005).
    [CrossRef]
  9. J. Ojeda-Castañeda, M. Martínez-Corral, P. Andrés, and A. Pons, “Strehl ratio versus defocus for noncentrally obscured pupils,” Appl. Opt. 33, 7611-7616 (1994).
    [CrossRef] [PubMed]
  10. S. Szapiel, “Maréchal intensity criteria modified for circular apertures with nonuniform intensity transmission: Dini series approach,” Opt. Lett. 2, 124-126 (1978).
    [CrossRef] [PubMed]
  11. W. B. King, “Dependence of the Strehl ratio on the magnitude of the variance of the wave aberration,” J. Opt. Soc. Am. 58, 655-661 (1968).
    [CrossRef]
  12. R. Herloski, “Strehl ratio for untruncated aberrated Gaussian beams,” J. Opt. Soc. Am. A 2, 1027-1029 (1985).
    [CrossRef]
  13. V. N. Mahajan, “Strehl ratio for primary aberrations in terms of their aberration variance,” J. Opt. Soc. Am. 73, 860-861 (1983).
    [CrossRef]
  14. M. J. YzuelJ. Campos, and F. Calvo, “Maréchal intensity criteria for apertures with polynomial non-uniform transmission. Evaluation of the diffraction focus and the Strehl ratio,” J. Mod. Opt. 38, 349-362 (1991).
    [CrossRef]
  15. H. T. Yura, “Variance of the Strehl ratio of an adaptive optics system,” J. Opt. Soc. Am. A 15, 2107-2110(1998).
    [CrossRef]
  16. V. H. Shui and N. A. Thyson, “Dependence of Strehl ratio on turbulence model,” Proc. SPIE 2222, 726-734(1994).
    [CrossRef]
  17. http://en.wikipedia.org/wiki/Expected_value.
  18. http://mathworld.wolfram.com/ExpectationValue.html.
  19. A. N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,” J. Fluid Mech. 13, 82-85 (1962).
    [CrossRef]

2005

2004

L. C. Roberts, Jr.,, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy “Is that really your Strehl ratio?” Proc. SPIE 5490, 504-515 (2004).

1998

1997

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1997), Chap. 9.1.3, Eq. (24), p. 464.

1994

1991

M. J. YzuelJ. Campos, and F. Calvo, “Maréchal intensity criteria for apertures with polynomial non-uniform transmission. Evaluation of the diffraction focus and the Strehl ratio,” J. Mod. Opt. 38, 349-362 (1991).
[CrossRef]

1985

1983

1982

1978

1974

1968

1962

A. N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,” J. Fluid Mech. 13, 82-85 (1962).
[CrossRef]

1947

A. Maréchal, “Étude des effets combinés de la diffraction et des aberrations géometriques sur l'image d'un point lumineux,” Rev. Opt. Theor. Instrum. 26,257-277 (1947)

1902

K. Strehl, “Über Luftschlieren und Zonenfehler,” Z. Instrumentenkd. 22, 213-217 (1902).

Andrés, P.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1997), Chap. 9.1.3, Eq. (24), p. 464.

Calvo, F.

M. J. YzuelJ. Campos, and F. Calvo, “Maréchal intensity criteria for apertures with polynomial non-uniform transmission. Evaluation of the diffraction focus and the Strehl ratio,” J. Mod. Opt. 38, 349-362 (1991).
[CrossRef]

Campos, J.

M. J. YzuelJ. Campos, and F. Calvo, “Maréchal intensity criteria for apertures with polynomial non-uniform transmission. Evaluation of the diffraction focus and the Strehl ratio,” J. Mod. Opt. 38, 349-362 (1991).
[CrossRef]

Christou, J. C.

L. C. Roberts, Jr.,, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy “Is that really your Strehl ratio?” Proc. SPIE 5490, 504-515 (2004).

Herloski, R.

King, W. B.

Kolmogorov, A. N.

A. N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,” J. Fluid Mech. 13, 82-85 (1962).
[CrossRef]

Lowenthal, D. D.

Macintosh, B. A.

L. C. Roberts, Jr.,, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy “Is that really your Strehl ratio?” Proc. SPIE 5490, 504-515 (2004).

Mahajan, V. N.

Marchis, F.

L. C. Roberts, Jr.,, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy “Is that really your Strehl ratio?” Proc. SPIE 5490, 504-515 (2004).

Maréchal, A.

A. Maréchal, “Étude des effets combinés de la diffraction et des aberrations géometriques sur l'image d'un point lumineux,” Rev. Opt. Theor. Instrum. 26,257-277 (1947)

Martial, G.

G. Martial, “Strehl ratio and aberration balancing,” J. Opt. Soc. Am. A 1, 164-170(1991); see comment surrounding Eq. (8).

Martínez-Corral, M.

Ojeda-Castañeda, J.

Perrin, M. D.

L. C. Roberts, Jr.,, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy “Is that really your Strehl ratio?” Proc. SPIE 5490, 504-515 (2004).

Pons, A.

Poyneer, L. A.

L. C. Roberts, Jr.,, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy “Is that really your Strehl ratio?” Proc. SPIE 5490, 504-515 (2004).

Roberts, L. C.

L. C. Roberts, Jr.,, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy “Is that really your Strehl ratio?” Proc. SPIE 5490, 504-515 (2004).

Shui, V. H.

V. H. Shui and N. A. Thyson, “Dependence of Strehl ratio on turbulence model,” Proc. SPIE 2222, 726-734(1994).
[CrossRef]

Sivaramakrishnan, A.

L. C. Roberts, Jr.,, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy “Is that really your Strehl ratio?” Proc. SPIE 5490, 504-515 (2004).

Strehl, K.

K. Strehl, “Über Luftschlieren und Zonenfehler,” Z. Instrumentenkd. 22, 213-217 (1902).

Szapiel, S.

Thyson, N. A.

V. H. Shui and N. A. Thyson, “Dependence of Strehl ratio on turbulence model,” Proc. SPIE 2222, 726-734(1994).
[CrossRef]

Troy, M.

L. C. Roberts, Jr.,, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy “Is that really your Strehl ratio?” Proc. SPIE 5490, 504-515 (2004).

van Dam, M. A.

L. C. Roberts, Jr.,, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy “Is that really your Strehl ratio?” Proc. SPIE 5490, 504-515 (2004).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1997), Chap. 9.1.3, Eq. (24), p. 464.

Yura, H. T.

Yzuel, M. J.

M. J. YzuelJ. Campos, and F. Calvo, “Maréchal intensity criteria for apertures with polynomial non-uniform transmission. Evaluation of the diffraction focus and the Strehl ratio,” J. Mod. Opt. 38, 349-362 (1991).
[CrossRef]

Appl. Opt.

J. Fluid Mech.

A. N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,” J. Fluid Mech. 13, 82-85 (1962).
[CrossRef]

J. Mod. Opt.

M. J. YzuelJ. Campos, and F. Calvo, “Maréchal intensity criteria for apertures with polynomial non-uniform transmission. Evaluation of the diffraction focus and the Strehl ratio,” J. Mod. Opt. 38, 349-362 (1991).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Lett.

Proc. SPIE

V. H. Shui and N. A. Thyson, “Dependence of Strehl ratio on turbulence model,” Proc. SPIE 2222, 726-734(1994).
[CrossRef]

L. C. Roberts, Jr.,, M. D. Perrin, F. Marchis, A. Sivaramakrishnan, R. B. Makidon, J. C. Christou, B. A. Macintosh, L. A. Poyneer, M. A. van Dam, and M. Troy “Is that really your Strehl ratio?” Proc. SPIE 5490, 504-515 (2004).

Rev. Opt. Theor. Instrum.

A. Maréchal, “Étude des effets combinés de la diffraction et des aberrations géometriques sur l'image d'un point lumineux,” Rev. Opt. Theor. Instrum. 26,257-277 (1947)

Z. Instrumentenkd.

K. Strehl, “Über Luftschlieren und Zonenfehler,” Z. Instrumentenkd. 22, 213-217 (1902).

Other

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1997), Chap. 9.1.3, Eq. (24), p. 464.

http://en.wikipedia.org/wiki/Expected_value.

http://mathworld.wolfram.com/ExpectationValue.html.

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Figures (4)

Fig. 1
Fig. 1

Strehl ratio for a Gaussian noise distribution.

Fig. 2
Fig. 2

Strehl ratio for a flat noise distribution.

Fig. 3
Fig. 3

Strehl ratio for a triangular noise distribution.

Fig. 4
Fig. 4

Strehl ratio for a hyperbolic secant noise distribution.

Equations (13)

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S ( σ ) e ( 2 π σ ) 2
S ( σ ) 1 ( 2 π σ ) 2 +
S = I σ ( 0 , 0 ) I ( 0 , 0 ) = | U ( x , y ) exp [ 2 π j Φ ( x , y ) ] exp [ 2 π j ( x ξ + y η ) ] d x d y | 2 | U ( x , y ) exp [ 2 π j ( x ξ + y η ) ] d x d y | 2 ] ξ = η = 0 .
S = | U ( x , y ) exp [ 2 π j Φ ( x , y ) ] d x d y | 2 | U ( x , y ) d x d y | 2 .
S = | U exp [ 2 π j Φ ( x , y ) ] d x d y | 2 | U d x d y | 2 = | 1 A exp [ 2 π j Φ ( x , y ) ] d x d y | 2 .
S = | 1 A 1 + 2 π j Φ ( x , y ) + 1 2 ( 2 π j Φ ( x , y ) ) 2 + d x d y | 2 .
Φ ¯ n = Φ n ( x , y ) d x d y d x d y ,
S | 1 A 1 + 2 π j Φ ( x , y ) + 1 2 ( 2 π j Φ ( x , y ) ) 2 + d x d y | 2 | 1 + 2 π j Φ ¯ 1 2 π 2 Φ ¯ 2 | 2 = ( 2 π Φ ¯ 1 ) 2 + ( 1 2 π 2 Φ ¯ 2 ) 2 4 π 2 Φ ¯ 1 2 + 1 4 π 2 Φ ¯ 2 + 1 4 π 2 ( Φ ¯ 2 Φ ¯ 1 2 ) + .
Δ Φ rms 2 = σ 2 = 1 A ( Φ Φ ¯ 1 ) 2 d A = 1 A ( Φ 2 d A 2 Φ ¯ 1 Φ d A Φ ¯ 1 2 d A ) = Φ ¯ 2 2 Φ ¯ 1 2 + Φ ¯ 1 2 = Φ ¯ 2 Φ ¯ 1 2 .
S ( σ ) 1 4 π 2 ( Φ ¯ 2 Φ ¯ 1 2 ) = 1 4 π 2 Δ Φ rms 2 = 1 4 π 2 σ 2 .
E ( g ( f ( ξ ) ) ) = g ( ξ ) PDF ( ξ ) d ξ .
S = | E ( exp [ 2 π j Φ ( x , y ) ] ) | 2 .
S ( σ ) = σ 2 | exp ( 2 π j ξ ^ σ ) PDF ( ξ ^ ) d ξ ^ | 2 = σ 2 | F [ PDF ( ξ ^ ) ] | 2 .

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